Question

Consider the following vector fields, where \(\mathbf{r}=\langle x, y, z\rangle\). a. Compute the curl of the field and verify that it has the same direction as the axis of rotation. b. Compute the magnitude of the curl of the field. $$\mathbf{F}=\langle 1,0,0\rangle \times \mathbf{r}$$

Short answer

Answer: The curl of the vector field is \(\nabla \times \mathbf{F} = \langle 0, 0, 0\rangle\), and its magnitude is 0.

Step-by-step solution

Compute Cross Product

First, compute the cross product of \(\langle 1, 0, 0\rangle\) and the vector \(\mathbf{r} = \langle x, y, z \rangle\): $$\mathbf{F}=\langle 1,0,0\rangle \times \langle x, y, z \rangle = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 0 \\ x & y & z \end{vmatrix} = \dots $$ Expanding the determinant along the first row, we get: $$\mathbf{F}=(0\cdot z - 0\cdot y)\mathbf{i} - (0\cdot z - 0\cdot x)\mathbf{j} + (0\cdot y - 0\cdot x)\mathbf{k} = \langle 0, 0, 0\rangle.$$

Compute the Curl

Now, compute the curl of the vector field: $$\nabla \times \mathbf{F} = \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\rangle \times \langle 0, 0, 0\rangle = \dots$$ Computing the cross product, we find: $$\nabla \times \mathbf{F} = \langle 0, 0, 0\rangle.$$

Verify the Direction

Since the curl of the vector field is \(\langle 0, 0, 0\rangle\), this means that the field has no rotation, and the direction of the curl coincides with the axis of rotation, as they are both at the origin.

Compute the Magnitude of the Curl

Compute the magnitude of the curl vector \(\nabla \times \mathbf{F} = \langle 0, 0, 0\rangle\): $$|\nabla \times \mathbf{F}| = \sqrt{(0^2)+(0^2)+(0^2)} = 0$$ The magnitude of the curl of the vector field is 0.

Key concepts

Cross Product

The cross product, also known as the vector product, is an operation performed on two vectors in three-dimensional space. It results in another vector that is perpendicular to the plane formed by the original two vectors. The cross product of vectors \textbf{A} and \textbf{B} is denoted as \textbf{A} \times \textbf{B}.

In the context of vector calculus, the cross product is used to compute quantities like torque, angular momentum, and the curl of a vector field. For vectors \textbf{A} = \textlangle a_1, a_2, a_3 \textrangle and \textbf{B} = \textlangle b_1, b_2, b_3 \textrangle, the cross product is a determinant of a matrix with the unit vectors \textbf{i}, \textbf{j}, and \textbf{k} as the first row, the components of \textbf{A} as the second row, and the components of \textbf{B} as the third row:\[\textbf{A} \times \textbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\]
A well-understood cross product is crucial for grasping the concept of the curl of a vector field, which represents the rotation or 'twist' of a field around a point.

Vector Calculus

Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields, often in three-dimensional Euclidean space. It is essential for understanding physical phenomena such as fluid flow and electromagnetic fields.

In vector calculus, several key operations are applied to vector fields, such as the gradient, divergence, and curl. The gradient measures the rate and direction of change in a scalar field, the divergence measures the magnitude of a source or sink at a given point in a vector field, and the curl measures the rotation of the field around a point.

The calculation of the curl involves both cross product and partial derivatives, which together help in determining the local rotation induced by the vector field at a point. For example, the curl of a velocity field in fluid dynamics gives us a sense of the whirlpools or eddies at specific locations within the fluid.

Determinants

Determinants are a mathematical tool used to solve systems of linear equations, evaluate certain matrix properties, and solve various vector problems. In the context of vector calculus, the determinant is used when computing the cross product.\[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\] here represents a 3 x 3 determinant that results in a vector perpendicular to both vectors \textbf{A} and \textbf{B}.

The value of the determinant is calculated by expanding along a row or a column according to specific rules, addressing the concept of cofactors and minors. In simple terms, the determinant provides the 'scaling factor' for transformations described by a matrix, which is why it's commonly associated with the area of parallelograms in a plane or the volume of parallelepipeds in three dimensions when working with vector cross products.

Partial Derivatives

Partial derivatives are derivatives of multivariable functions with respect to one variable, holding the other variables constant. They are a fundamental tool in multivariable calculus, which includes vector calculus. They express the rate at which a function changes as one of the variables changes, while the other variables are held fixed.

In the computation of the curl of a vector field, partial derivatives play a critical role. Given a vector field \textbf{F} = \textlangle F_x, F_y, F_z \textrangle, the curl of \textbf{F} is a vector field computed as:\[abla \times \textbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\]
Each component of the curl is determined by the difference in the relevant partial derivatives, creating a new vector that describes the rotational effect at each point within the vector field. This derivation highlights the interaction between calculus and algebra when analyzing the properties of a vector field.